# Five balls of different colors are to placed in there boxes of different size. Each box can hold all five. In how many different ways can we place the balls so that no box remains empty?

Navjyot Kalra
8 years ago
Hello Student,
The various possibilities to put 5 different balls in 3 different size boxes, when no box remains empty : The balls can be 1, 1 and 3 in different or 2, 2, 1.
Case I : To put 1, 1 and 3 balls in different boxes. Selection of 1, 1 and 3 balls out of 5 balls can be done in 5 C1 x 4 C1 x 3 C3 ways and then 1, 1, 3 can permute (as different size boxes) in 3 ! ways.
∴ No. of ways
= 5 C1 x 4 C1 x 3 C3 x 3! = 5 x 4 x 1 x 6 = 120
Case II : To put 2, 2 and 1 ball in different boxes. Selection of 2, 2 and 1 balls out of 5 balls can be done in 5 C2 x 3 C2 x 1 C1 ways
And then 2, 2, 1 can permute (different boxes) in 3! Ways
∴ No. of ways
5 C1 x 3 C2 x 1 C1 x 3! = 10 x 3 x 1 x 6 = 180
Combining case I and II, total number of required ways are
= 120 + 180 = 300.

Thanks
navjot kalra